Convergence of Some Convex Message Passing Algorithms to a Fixed Point

A popular approach to the MAP inference problem in graphical models is to minimize an upper bound obtained from a dual linear programming or Lagrangian relaxation by (block-)coordinate descent. Examples of such algorithms are max-sum diffusion and sequential tree-reweighted message passing. Convergence properties of these methods are currently not fully understood. They have been proved to converge to the set characterized by local consistency of active constraints, with unknown convergence rate; however, it was not clear if the iterates converge at all (to any single point). We prove a stronger result (which was conjectured before but never proved): the iterates converge to a fixed point of the algorithm. Moreover, we show that they achieve precision $\varepsilon>0$ in $\mathcal{O}(1/\varepsilon)$ iterations. We first prove this for a version of coordinate descent applied to a general piecewise-affine convex objective, using a novel proof technique. Then we demonstrate the generality of this approach by reducing some popular coordinate-descent algorithms to this problem. Finally we show that, in contrast to our main result, a similar version of coordinate descent applied to a constrained optimization problem need not converge.

On Coordinate Minimization of Convex Piecewise-Affine Functions

A popular class of algorithms to optimize the dual LP relaxation of the discrete energy minimization problem (a.k.a.\ MAP inference in graphical models or valued constraint satisfaction) are convergent message-passing algorithms, such as max-sum diffusion, TRW-S, MPLP and SRMP. These algorithms are successful in practice, despite the fact that they are a version of coordinate minimization applied to a convex piecewise-affine function, which is not guaranteed to converge to a global minimizer. These algorithms converge only to a local minimizer, characterized by local consistency known from constraint programming. We generalize max-sum diffusion to a version of coordinate minimization applicable to an arbitrary convex piecewise-affine function, which converges to a local consistency condition. This condition can be seen as the sign relaxation of the global optimality condition.

Primal View on Belief Propagation

It is known that fixed points of loopy belief propagation (BP) correspond to stationary points of the Bethe variational problem, where we minimize the Bethe free energy subject to normalization and marginalization constraints. Unfortunately, this does not entirely explain BP because BP is a dual rather than primal algorithm to solve the Bethe variational problem -- beliefs are infeasible before convergence. Thus, we have no better understanding of BP than as an algorithm to seek for a common zero of a system of non-linear functions, not explicitly related to each other. In this theoretical paper, we show that these functions are in fact explicitly related -- they are the partial derivatives of a single function of reparameterizations. That means, BP seeks for a stationary point of a single function, without any constraints. This function has a very natural form: it is a linear combination of local log-partition functions, exactly as the Bethe entropy is the same linear combination of local entropies.

Zero-Temperature Limit of a Convergent Algorithm to Minimize the Bethe Free Energy

After the discovery that fixed points of loopy belief propagation coincide with stationary points of the Bethe free energy, several researchers proposed provably convergent algorithms to directly minimize the Bethe free energy. These algorithms were formulated only for non-zero temperature (thus finding fixed points of the sum-product algorithm) and their possible extension to zero temperature is not obvious. We present the zero-temperature limit of the double-loop algorithm by Heskes, which converges a max-product fixed point. The inner loop of this algorithm is max-sum diffusion. Under certain conditions, the algorithm combines the complementary advantages of the max-product belief propagation and max-sum diffusion (LP relaxation): it yields good approximation of both ground states and max-marginals.